Principal component filter banks: existence issues, and application to modulated filter banks

Principal component filter banks (PCFBs) sequentially compress most of the input signal energy into the first few subbands, and are mathematically defined using the notion of majorization. In a series of recent works, we have exploited connections between majorization and convexity theory to provide a unified explanation of PCFB optimality for numerous signal processing problems, involving compression, noise suppression and progressive transmission, However PCFBs are known to exist for all input spectra only for three special classes of orthonormal filter banks (FBs): any class of two channel FBs, the transform coder class and the unconstrained class. This paper uses the developed theory to describe techniques to examine existence of PCFBs. We prove that the classes of DFT and cosine-modulated FBs do not have PCFBs for large families of input spectra. This result is new and quite different from most known facts on nonexistence of PCFBs, which usually involve very specific examples and proofs with numerical optimizations